Genealogy with seasonality, the basic reproduction number, and the influenza pandemic

Journal of Mathematical Biology

Volumn 62, Issue 5, 2011, Pages 741-762

  Bacaër, Nicolas ^a   Dads, El Hadi Ait ^b  

^a Sorbonne Universités   (France)

^b CADI AYYAD UNIVERSITY   (Morocco)

Author keywords

Basic reproduction number; Epidemics; H1N1 influenza; Seasonality

Indexed keywords

ALGORITHM; ARTICLE; BASIC REPRODUCTION NUMBER; BIOLOGICAL MODEL; DISEASE TRANSMISSION; HUMAN; INCIDENCE; INFLUENZA; INFLUENZA VIRUS A H1N1; PANDEMIC; SEASON;

ALGORITHMS; BASIC REPRODUCTION NUMBER; HUMANS; INCIDENCE; INFLUENZA A VIRUS, H1N1 SUBTYPE; INFLUENZA, HUMAN; MODELS, BIOLOGICAL; PANDEMICS; SEASONS;

EID: 79953038195     PISSN: 03036812     EISSN: 14321416     Source Type: Journal    
DOI: 10.1007/s00285-010-0354-8     Document Type: Article

References (44)

1. Aronsson G, Kellogg RB (1978) On a differential equation arising from compartmental analysis. Math Biosci 38: 113-122.
2. Bacaër N, Guernaoui S (2006) The epidemic threshold of vector-borne diseases with seasonality. J Math Biol 53: 421-436.
3. 0 for vector-borne diseases with a periodic vector population. Bull Math Biol 69: 1067-1091.
4. Bacaër N, Ouifki R (2007) Growth rate and basic reproduction number for population models with a simple periodic factor. Math Biosci 210: 647-658.
5. Bacaër N, Abdurahman X (2008) Resonance of the epidemic threshold in a periodic environment. J Math Biol 57: 649-673.
6. Bacaër N, Gomes MGM (2009) On the final size of epidemics with seasonality. Bull Math Biol 71: 1954-1966.
7. Bacaër N (2009) Periodic matrix population models: growth rate, basic reproduction number, and entropy. Bull Math Biol 71: 1781-1792.
8. Bapat RB, Raghavan TES (1997) Nonnegative matrices and applications. Cambridge University Press, London.
9. Berman A, Plemmons RJ (1979) Nonnegative matrices in the mathematical sciences. Academic Press, New York.
10. Boëlle PY, Bernillon P, Desenclos JC (2009) A preliminary estimation of the reproduction ratio for new influenza A (H1N1) from the outbreak in Mexico, March-April 2009. Euro Surveill 14(19): pii=19205.
11. Burlando L (1991) Monotonicity of spectral radius for positive operators on ordered Banach spaces. Arch Math 56: 49-57.
12. Cauchemez S, Valleron AJ, Boelle PY, Flahault A, Ferguson NM (2008) Estimating the impact of school closure on influenza transmission from Sentinel data. Nature 452(7188): 750-754.
13. Coale AJ (1970) The use of Fourier analysis to express the relation between time variations in fertility and the time sequence of births in a closed human population. Demography 7: 93-120.
14. Coale AJ (1972) The growth and structure of human populations, a mathematical investigation. Princeton University Press, Princeton.
15. Degla G (2008) An overview of semi-continuity results on the spectral radius and positivity. J Math Anal Appl 338: 101-110.
16. Diekmann O, Heesterbeek JAP (2000) Mathematical epidemiology of infectious diseases. Wiley, Chichester.
17. Drábek P, Milota J (2007) Methods of nonlinear analysis applications to differential equations. Birkhäuser, Basel.
18. Drnovšek R (2000) Bounds for the spectral radius of positive operators. Comment Math Univ Carol 41: 459-467.
19. Dushoff J, Plotkin JB, Levin SA, Earn DJD (2004) Dynamical resonance can account for seasonality of influenza epidemics. Proc Natl Acad Sci USA 101: 16915-16916.
20. Ediev D (2003) On monotone convergence to stability. Demogr Res 8: 31-60.
21. Feller W (1941) On the integral equation of renewal theory. Ann Math Stat 12: 243-267.
22. Fraser C, Donnelly CA, Cauchemez S et al (2009) Pandemic potential of a strain of influenza A (H1N1): early findings. Science 324: 1557-1561.
23. Grassly NC, Fraser C (2006) Seasonal infectious disease epidemiology. Proc R Soc B 273: 2541-2550.
24. Heesterbeek JAP, Roberts MG (1995) Threshold quantities for helminth infections. J Math Biol 33: 415-434.
25. Inaba H, Nishiura H (2008) The basic reproduction number of an infectious disease in a stable population: the impact of population growth rate on the eradication threshold. Math Model Nat Phenom 3(7): 194-228.
26. Jagers P, Nerman O (1985) Branching processes in periodically varying environment. Ann Probab 13: 254-268.
27. Kato T (1982) Superconvexity of the spectral radius, and convexity of the spectral bound and the type. Math Z 180: 265-273.
28. Kermack WO, McKendrick AG (1927) A contribution to the mathematical theory of epidemics. Proc R Soc A 115: 700-721.
29. Kress R (1999) Linear integral equations, 2nd edn. Springer, New York.
30. Lipsitch M, Viboud C (2009) Influenza seasonality: lifting the fog. Proc Natl Acad Sci USA 106: 3645-3646.
31. Lotka AJ (1939) Théorie analytique des associations biologiques, 2e partie. Hermann, Paris.
32. Michel P, Mischler S, Perthame B (2005) General relative entropy inequality: an illustration on growth models. J Math Pures Appl 84: 1235-1260.
33. Munayco CV, Gomez J, Laguna-Torres VA et al (2009) Epidemiological and transmissibility analysis of influenza A(H1N1)v in a southern hemisphere setting: Peru. Euro Surveill 14(32 pii): 19299.
34. Nakata Y, Kuniya T (2010) Global dynamics of a class of SEIRS epidemic models in a periodic environment. J Math Anal Appl 363: 230-237.
35. Nishiura H, Castillo-Chavez C, Safan M, Chowell G (2009) Transmission potential of the new influenza A (H1N1) virus and its age-specificity in Japan. Euro Surveill 14(22): pii-19227.
36. Perthame B (2007) Transport equations in biology. Birhäuser, Basel.
37. Pourbohloul B, Ahued A, Davoudi B et al (2009) Initial human transmission dynamics of the pandemic (H1N1) 2009 virus in North America. Influenza Other Respi Viruses 3(5): 215-222.
38. Shaman J, Kohn M (2009) Absolute humidity modulates influenza survival, transmission, and seasonality. Proc Natl Acad Sci USA 106: 3243-3248.
39. Thieme HR (1984) Renewal theorems for linear periodic Volterra integral equations. J Integral Equ 7: 253-277.
40. Thieme HR (1998) Remarks on resolvent positive operators and their perturbation. Discret Contin Dyn Syst 4: 73-90.
41. Thieme HR (2009) Spectral bound and reproduction number for infinite population structure and time-heterogeneity. SIAM J Appl Math 70: 188-211.
42. Wang W, Zhao XQ (2008) Threshold dynamics for compartmental epidemic models in periodic environments. J Dyn Differ Equ 20: 699-717.
43. Wesley CL, Allen LJS (2009) The basic reproduction number in epidemic models with periodic demographics. J Biol Dyn 3: 116-129.
44. Williams BG, Dye C (1997) Infectious disease persistence when transmission varies seasonally. Math Biosci 145: 77-88.